Theorem eqeqan12d 2368

Description: A useful inference for substituting definitions into an equality. (Contributed by NM, 9-Aug-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Hypotheses

Ref	Expression
eqeqan12d.1	⊢ (φ → A = B)
eqeqan12d.2	⊢ (ψ → C = D)

Assertion

Ref	Expression
eqeqan12d	⊢ ((φ ∧ ψ) → (A = C ↔ B = D))

Proof of Theorem eqeqan12d

Step	Hyp	Ref	Expression
1		eqeqan12d.1	. 2 ⊢ (φ → A = B)
2		eqeqan12d.2	. 2 ⊢ (ψ → C = D)
3		eqeq12 2365	. 2 ⊢ ((A = B ∧ C = D) → (A = C ↔ B = D))
4	1, 2, 3	syl2an 463	1 ⊢ ((φ ∧ ψ) → (A = C ↔ B = D))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-cleq 2346

This theorem is referenced by:  eqeqan12rd  2369  adj11  3890  pw1equn  4332  pw1eqadj  4333  eqfnfv2  5394  pw1fnf1o  5856  dflec2  6211  tc11  6229